Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 087, 39 pages      arXiv:1104.2459
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

Martijn Caspers
Radboud Universiteit Nijmegen, IMAPP, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Received April 14, 2011, in final form August 30, 2011; Published online September 06, 2011

We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in SL(2,C) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a Z2-grading, we obtain product formulae for little q-Jacobi functions.

Key words: locally compact quantum groups; Plancherel theorem; Fourier transform; spherical functions.

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  1. Caspers M., The Lp-Fourier transform on locally compact quantum groups, J. Operator Theory, to appear, arXiv:1008.2603.
  2. Caspers M., Koelink E., Modular properties of matrix coefficients of corepresentations of a locally compact quantum group, J. Lie Theory 21 (2011), 905-928, arXiv:1003.2278.
  3. De Commer K., On a correspondence between quantum SU(2), quantum E(2) and extended quantum SU(1,1), Comm. Math. Phys. 304 (2011), 187-228, arXiv:1004.4307.
  4. Desmedt P., Aspects of the theory of locally compact quantum groups: Amenability-Plancherel measure, Ph.D. Thesis, Katholieke Universiteit Leuven, 2003.
  5. van Dijk G., Introduction to harmonic analysis and generalized Gelfand pairs, de Gruyter Studies in Mathematics, Vol. 36, Walter de Gruyter & Co., Berlin, 2009.
  6. Dixmier J., Les algèbres d'opérateurs dans l'espace hilbertien, Gauthiers-Villars, Paris, 1957.
  7. Dixmier J., Les C*-algèbres et leurs représentations, Gauthiers-Villars, Paris, 1969.
  8. Faraut J., Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, in Analyse Harmonique (Universite de Nancy I, 1980), Editors J.L. Clerc et al., Les Cours du C.I.M.P.A., Centre International de Mathematiques Pures et Appliquees, Nice, France, 1983, 315-446.
  9. Floris P.G.A., Gel'fand pair criteria for compact matrix quantum groups, Indag. Math. (N.S.) 6 (1995), 83-98.
  10. Groenevelt W., Koelink E., Kustermans J., The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C), Int. Math. Res. Not. 2010 (2010), no. 7, 1167-1314, arXiv:0905.2830.
  11. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras, Vol. II, Advanced theory, Graduate Studies in Mathematics, Vol. 16, American Mathematical Society, Providence, RI, 1997.
  12. Kasprzak P., On a certain approach to quantum homogeneous spaces, arXiv:1007.2438.
  13. Koelink E., Kustermans J., A locally compact quantum group analogue of the normalizer of SU(1,1) in SL(2,C), Comm. Math. Phys. 233 (2003), 231-296, math.QA/0105117.
  14. Koelink E., Stokman J.V., Fourier transforms on the quantum SU(1,1) group, With an appendix by Mizan Rahman, Publ. Res. Inst. Math. Sci. 37 (2001), 621-715, math.QA/9911163.
  15. Koelink H.T., Askey-Wilson polynomials and the quantum SU(2) group: survey and applications, Acta Appl. Math. 44 (1996), 295-352.
  16. Koelink H.T., The quantum group of plane motions and the Hahn-Exton q-Bessel function, Duke Math. J. 76 (1994), 483-508.
  17. Koornwinder T.H., Positive convolution structures associated with quantum groups, in Probability Measures on Groups, X (Oberwolfach, 1990), Plenum, New York, 1991, 249-268.
  18. Kustermans J., Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001), 289-338, math.OA/9902015.
  19. Kustermans J., Locally compact quantum groups, in Quantum Independent Increment Processes. I, Lecture Notes in Math., Vol. 1865, Springer, Berlin, 2005, 99-180.
  20. Kustermans J., One-parameter representations of C*-algebras, funct-an/9707009.
  21. Kustermans J., Vaes S., Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), 837-934.
  22. Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68-92, math.OA/0005219.
  23. Lance C., Direct integrals of left Hilbert algebras, Math. Ann. 216 (1975), 11-28.
  24. Noumi M., Yamada H., Mimachi K., Finite-dimensional representations of the quantum group GLq(n,C) and the zonal spherical functions on Uq(n−1)\Uq(n), Japan. J. Math. (N.S.) 19 (1993), 31-80.
  25. Nussbaum A.E., Reduction theory for unbounded closed operators in Hilbert space, Duke Math. J. 31 (1964), 33-44.
  26. Podkolzin G.B., Vainerman L.I., Quantum Stiefel manifold and double cosets of quantum unitary group, Pacific J. Math. 188 (1999), 179-199.
  27. Raeburn I., Williams D.P., Morita equivalence and continuous-trace C*-algebras, Mathematical Surveys and Monographs, Vol. 60, American Mathematical Society, Providence, RI, 1998.
  28. Salmi P., Skalski A., Idempotent states on locally compact quantum groups, Quart. J. Math., to appear, arXiv:1102.2051.
  29. Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York - Heidelberg, 1979.
  30. Takesaki M., Theory of operator algebras. II, Springer-Verlag, Berlin, 2003.
  31. Timmermann T., An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.
  32. Tomatsu R., A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys. 275 (2007), 271-296, math.OA/0611327.
  33. Vaes S., A new approach to induction and imprimitivity results, J. Funct. Anal. 229 (2005), 317-374, math.OA/0407525.
  34. Vaes S., A Radon-Nikodym theorem for von Neumann algebras, J. Operator Theory 46 (2001), 477-489, math.OA/9811122.
  35. Vaes S., The unitary implementation of a locally compact quantum group action, J. Funct. Anal. 180 (2001), 426-480, math.OA/0005262.
  36. Vaes S., Vainerman L.I., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math. 175 (2003), 1-101, math.OA/0101133.
  37. Vaes S., Vainerman L.I., On low-dimensional locally compact quantum groups, in Locally Compact Quantum Groups and Groupoids (Strasbourg, 2002), IRMA Lect. Math. Theor. Phys., Vol. 2, de Gruyter, Berlin, 2003, 127-187, math.QA/0207271.
  38. Vainerman L.I., Gel'fand pair associated with the quantum group of motions of the plane and q-Bessel functions, Rep. Math. Phys. 35 (1995), 303-326.
  39. Vainerman L.I., Gel'fand pairs of quantum groups, hypergroups and q-special functions, in Applications of Hypergroups and Related Measure Algebras (Seattle, WA, 1993), Contemp. Math., Vol. 183, Amer. Math. Soc., Providence, RI, 1995, 373-394.
  40. Vainerman L.I., Hypergroup structures associated with Gel'fand pairs of compact quantum groups, in Recent Advances in Operator Algebras (Orléans, 1992), Asterisque (1995), no. 232, 231-242.
  41. Van Daele A., The Fourier transform in quantum group theory, in New Techniques in Hopf Algebras and Graded Ring Theory, K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 2007, 187-196, math.RA/0609502.
  42. Van Daele A., Locally compact quantum groups. A von Neumann algebra approach, math.OA/0602212.
  43. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions, Vol. 1, Simplest Lie groups, special functions and integral transforms, Kluwer Academic Publishers Group, 1991.
  44. Vogan D.A., Representations of real reductive Lie groups, Progress in Mathematics, Vol. 15, Birkhäuser, Boston, Mass., 1981.

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